EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2613-2631 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Different Types of Mappings in Fuzzy Bitopological Spaces Ahlam Ahmed Alharbi1,2, Adem Kilicman2,∗ 1 Department of Mathematics, Faculty of Science, Taibah University, Madinah 42353, Kingdom of Saudi Arabia 2 Department of Mathematics, Faculty of Science University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia Abstract. The principal objective of this research is to present generalized function ideas includ- ing: fuzzy generalized continuity, generalized strong continuity, generalized irresoluteness, gener- alized open and closed mappings. The last part of our study focuses on homomorphisms in fuzzy bitopological spaces. We also explore the relationships between these concepts, their characteris- tics, compositions, and important theories, along with some relevant counterexamples. 2020 Mathematics Subject Classifications: 54A40, 03E72, 47S40, 94D05, 03B52, 28E10 Key Words and Phrases: Fuzzy bitopology space (fbts), fuzzy generalized closed groups ((i, j) − gψ − cld), fuzzy generalized continuous ((i, j) − gψ − conts), fuzzy generalized irreso- lute ((i, j)− gψ− irresolute), fuzzy generalized strongly continuous ((i, j)− gψ− strongly conts), fuzzy generalized open mapping ((i, j) − gψ − open mapping), fuzzy generalized closed mapping ((i, j)− gψ − closed mapping). 1. Introduction Our focus in this study is on fuzzy bitopology filed, that was developed of fuzzy topol- ogy and presented for the first time in 1965 by scientist Zadeh [14]. After that, some scientists developed the concepts of fuzzy topology by adapting fundamental ideas from general topology to fuzzy topology. For example, in 1968 Chang created several fuzzy concepts [8]. Then, fuzzy bitopological spaces were introduced by Kandil in 1989[4]. And hence, Balasubramanian and Sundaram created generalized fuzzy closed groups in fuzzy topology space in 1997 [11]. In addition, some scientists presented several studies on generalized closed group in fuzzy space [16, 18, 24]. Many studies about mappings in general topological space have been made by scientists, including [6, 12, 23]. As are some scholars also presented different types of studies on functions in fuzzy topology space [5, 7, 10, 13, 15, 17]. Also, there has been research that showed several mapping forms, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4972 Email addresses: aasehli@taibahu.edu.sa (A. A. Alharbi), akilic@upm.edu.my (A. Kilicman) https://www.ejpam.com 2613 © 2023 EJPAM All rights reserved. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2614 including irresolute and strong functions, such as [20–22]. There is also a lot of research related to the topic of this manuscript but with practical application in real-life scenarios, such as ”A comparison of three types of rough fuzzy sets based on two universal sets” [1]. Also, ”On Fuzzy Point Applications of Fuzzy Topological Spaces” [2]. Since, generalized closed sets in fuzzy bitopology spaces are essential for incorporating flexibility, granularity, and uncertainty in the study of fuzzy sets and various topological properties, such as con- tinuity. This use has several advantages, it gives a more precise description of the behavior of fuzzy functions. So, this study aims to introduce and explore a range of generalized function concepts within the framework of fuzzy bitopological spaces. More precisely, in the concepts of fuzzy generalized continuity, generalized strong continuity, generalized ir- resoluteness, as well as generalized open and closed mappings. Furthermore, we investigate the concept of homomorphism in the context of fuzzy bitopological spaces. Our study not only introduces these concepts but also delves into their interrelationships, characteristics, composition, and highlights important theories and counterexamples. Through this com- prehensive review, we contribute to a deeper understanding of these fundamental concepts and their applications in the context of fuzzy bitopological spaces. The study is set up as follows: The history, importance, and related research of the subject are examined in section 1 (introduction). In section 2 (preliminaries), we outline a few key antecedent ideas that are relevant to our research. The concept of generalized continuous concepts is presented in section 3 (Types of Fuzzy Generalized Continuous Mappings), which also discusses them in connection to important theorems and distinctive characteristics. How- ever, the types of strong continuity and irresoluteness functions are defined in section 4 (Types of Fuzzy Generalized Strongly Continuity and Irresolute Mapping), which also examines how these definitions relate to the major theories and some significant exam- ples. We also provided crucial definitions and theorems for open and closed mappings in section 5 (Types of Fuzzy Generalized Open and Closed Mappings). In section 6 (Fuzzy Generalized Homomorphism Mapping) we have presented a definition of homomorphism and reviewed the main theories and their relationship to the above functions. Finally, in section 7 (Conclusion), we compile our findings. 2. Preliminaries In the next section, we mention a few previous concepts which are fundamental to this study. Definition 1. [19] Assume I stands for the unit period [0, 1] and X is not a blank, then: (1) a fuzzy set M is referred to a function with domain X and range I, M(t) ∈ (0, 1] if t ∈M , as M(t) = 0 when t ̸∈M . (2) M is included in L as shown by M ⊆ L if M(t) ≤ L(t), while t ∈ X (3) M ∨ L is the combination of groups defined as (M ∨ L)(t) = upper{M(t), L(t)} ∀ t ∈ X. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2615 (4) M ∧ L is the intersection that defiend as (M ∧ L)(t) = lower{M(t), L(t)} ∀ t ∈ X. (5) M c is the completeness that defiend as (M(t))c = 1−M(t), ∀ t ∈ X. The concepts of fuzzy topology as well as fuzzy bitopological spaces are shown below: Definition 2. [19] The pair (X, δ) is consider fuzzy topology if the next three conditions holds: 1. 0, 1 ∈ δ, since 0(t) = 0, 1(t) = 1, as t ∈ X. 2. M ∧ L ∈ δ, ∀M,L ∈ δ,. 3. ∨i∈IMi ∈ δ, ∀(Mi∈I) ∈ δ,. The pair (X, δ) is referred to as ”fuzzy topology space,” or ”fts” shortly. Also, the parts of δ are known as open fuzzy groups. When F ∈ δ, thus F c is regarded as closed fuzzy group, and the set of all closed fuzzy groups denoted by Fδ. Definition 3. [4] A bitopology fuzzy spaces, often known as fbts, (X, δ1, δ2) as X is not empty, and δ1, δ2 are fuzzy topological spaces on X. during the course of this research, X conducts fuzzy bitopology (X, δ1, δ2) and Y takes (Y, σ1, σ2) so that i ̸= j, as i, j ∈ {1, 2} Definition 4. [9]. A fuzzy group µ of X is referred to fuzzy point (singleton) iff µ(t) = r, (0 < r ≤ 1) with a specific t ∈ X, µ(h) = 0 with each elements h of X excluding t, and it is indicated by tr. Sometimes we refer to tr as a fuzzy point if 0 < r < 1. Additionally, S(X) refers to the set of each fuzzy points (singletons) included in X. One of the fundamental ideas is the continuous and irresolute mapping, which were defined as: Definition 5. [19] If t is a function from (X, δ) to (Y, σ).Then t is fuzzy δ−continuous iff t−1(W ) ∈ δ, ∀W ∈ σ. Definition 6. [23] A function t : (X, δ) −→ (Y, σ) is known as fuzzy α− irresolute when t−1(W ) is fuzzy α−open of X on all fuzzy α−open W of Y . One of the fundamental ideas in the research is the generalized fuzzy closed group, that is known as follows: Definition 7. [11] K is named generalized fuzzy closed if closure K is subgroup of R, as K is subgroup of R, which is fuzzy open. i.e., K is generalized fuzzy closed when cl(K) ≤ R, whatever K ≤ R, R is fuzzy open. In the following sections, we divided the work into four parts: fuzzy (i, j)− generalized ψ continuity, (i, j)− generalized ψ strongly continuity and irresolute, (i, j)−generalized ψ open and closed mapping and last part is the fuzzy homomorphism. Also, we apply some theorems, some corollaries. As it includes important examples and diagrams to explain the relations via instructors. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2616 3. Types of Fuzzy Generalized Continuous Mappings We define and investigate some concepts of fuzzy generalized continuous mapping which includes fuzzy (i, j)−gα−conts, (i, j)−gs−conts, (i, j)−gp−conts, (i, j)−gβ−conts, and we denote for them by (i, j)− gψ−conts. Definition 8. Any subgroup K of fbts (X, δ1, δ2) is named as: (1) (i, j)−generalized ψ−closed (simply, (i, j)−gψ− cld) when δj−ψ− cl(K) ≤W , while K ≤ W , W ∈ δi, as ψ containing the kinds (alpha (α), semi (s), pre (p), and beta (β)). (2) (i, j)− gψ − open is the complement of the group (i, j)− gψ − cld. Remark 1. (1) A class of each fuzzy (i, j) − gψ−open, (i, j) − gψ − cld of (X, δ1, δ2) is represented by Ofgψ (i,j), F fgψ (i,j), and so forth. (2) The class of each gψ−open, gψ − cld subgroups of X in relation to δi represented by Ofgψ i , and Ffgψ i , i = 1, 2. In the following, we introduce the most important definitions and theories of the con- cept of generalized continuous: Definition 9. A function t : (X, δ1, δ2) → (Y, σ1, σ2) is named fuzzy (i, j)−generalized ψ− continuous (briefly, (i, j) − gψ − conts) when the opposite image of all fuzzy open group of (Y, σj) is fuzzy (i, j)− gψ − open group of (X, δ1, δ2). By using the complement of the above definition we get the coming remark: Remark 2. (i) Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Hence t is fuzzy (i, j)− gψ− conts iff ∀ fuzzy closed group V of (Y, σj), t −1(V ) is fuzzy (i, j)− gψ − cld group of X. (ii) By setting δi = δj, σi = σj in Definition 9, we find any fuzzy (i, j) − gψ − conts is fuzzy gψ − conts. Theorem 1. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j) − gψ − conts. Then any fuzzy point xr in X with σj −Q− nbd H of t(xr), ∃ fuzzy (i, j)− gψ −Q− nbd R of xr as t(R) ≤ H. Proof. Let xr ∈ IX and H ∈ NQ j (t(xr)). Then ∃ W ∈ σj as t(xr) qW ≤ H, and hence t−1(W ) is fuzzy (i, j) − gψ − open in X with xr q t −1(W ) ≤ t−1(H). If we take t−1(W ) = R, then ∃R ∈ NgψQ (i,j) (xr), as R ≤ t−1(H). So t(R) ≤ H. By using the relations via Ngψ (i,j), N gψQ (i,j) in [3], and the above theorem we get the next corollary: Corollary 1. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) be fuzzy (i, j)− gψ − conts. Then ∀ xr ∈ IX and ∀ H ∈ Nj(t(xr)), ∃ R ∈ Ngψ (i,j)(xr) as t(R) ≤ H. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2617 Theorem 2. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Then the coming claims are hold: (1) If t is fuzzy (i, j)− g − conts, hence it is (i, j)− gα− conts (2) If t is fuzzy (i, j)− gα− conts, hence it is (i, j)− gp− conts also (i, j)− gs− conts. (3) If t is fuzzy (i, j)− gp− conts or (i, j)− gs− conts, hence it is (i, j)− gβ − conts. Proof. It is clear from Definition 9, relationships via each kinds of fuzzy (i, j) general- ized neigborhoods in the refrence [3]. Remark 3. The following diagram explain the relation between statements in the above theorem. Figure 1: Presents the relationships via all varieties of fuzzy (i, j)− gψ − conts. The examples below demonstrate that the reverse implications of Figure (1) are gen- erally not true: Example 1. Suppose E,F , and G are fuzzy subgroups of X = {a, b}. We determine them as: E(a, b) = {0.5, 0.4}, F (a, b) = {0.7, 0.5}, and G(a, b) = {0.4, 0.4}. Consider the fuzzy bitopology δ1 = {0, 1, E} also δ2 = {0, 1, F,G} on X. Suppose N with M are fuzzy subgroups of Y = {r, h} defined as: N(r, h) = {0.2, 0.5}, M(r, h) = {0.7, 0.6}. Consider the fuzzy bitopology σ1 = {0, 1, N} with σ2 = {0, 1,M} on Y and t(a) = r, t(b) = h. One may notice that t is fuzzy (1, 2) − gα − conts, but not fuzzy (1, 2) − g − conts as t−1(M c) ≤ E ∈ δ1 but δ2 − cl(t−1(M c)) ≰ E. The next example clear that (1, 2)− gp− conts ⇏ (1, 2)− gα− conts. Example 2. Suppose E,F , and G are fuzzy subgroups of X = {a, b}. We determine them as: E(a, b) = {0.7, 0.5}, F (a, b) = {0.6, 0.8}, and G(a, b) = {0.4, 0.3}. Consider the fuzzy bitopology δ1 = {0, 1, E} also δ2 = {0, 1, F,G} on X. Suppose N with M are fuzzy subgroups of Y = {r, h} defined as: N(r, h) = {0.2, 0.5}, M(r, h) = {0.8, 0.6}. Consider the fuzzy bitopology σ1 = {0, 1, N} with σ2 = {0, 1,M} on Y and t(a) = r, t(b) = h. One may notice that t is fuzzy (1, 2) − gp − conts, but not fuzzy (1, 2) − gα − conts as t−1(M c) ≤ E ∈ δ1 but δ2 − α− cl(t−1(M c)) ≰ E. The next example proves (1, 2)− gs− conts ⇏ (1, 2)− gα− conts. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2618 Example 3. Suppose E,F,G, and H are fuzzy subgroups of X = {a, b}. We determine them as: E(a, b) = {0.7, 0.5} F (a, b) = {0.5, 0.4}, G(a, b) = {0.4, 0.3}, and H(a, b) = {0.5, 0.6}. Consider the fuzzy bitopology δ1 = {0, 1, E} with δ2 = {0, 1, F,G} on X. Suppose N with M are fuzzy subgroups of Y = {r, h} defined as: N(r, h) = {0.2, 0.5}, M(r, h) = {0.5, 0.5}. Consider the fuzzy bitopology σ1 = {0, 1, N} also σ2 = {0, 1,M} on Y and t(a) = r, t(b) = h. One may notice that t is fuzzy (1, 2)− gs− conts, but not fuzzy (1, 2)− gα− conts since t−1(M c) ≤ E ∈ δ1 but δ2 − α− cl(t−1(M c)) ≰ E. The following example clear that (1, 2)− gβ − conts ⇏ (1, 2)− gs− conts. Example 4. Suppose E,F,G, and H are fuzzy subgroups of X = {a, b}. We determine them as: E(a, b) = {0.5, 0.7} F (a, b) = {0.6, 0.5}, G(a, b) = {0.4, 0.3}, and H(a, b) = {0.6, 0.5}. Consider the fuzzy bitopology δ1 = {0, 1, E}, δ2 = {0, 1, F,G} on X. Suppose N , M are fuzzy subgroups of Y = {r, h} defined as: N(r, h) = {0.2, 0.5}, M(r, h) = {0.5, 0.5}. Consider the fuzzy bitopology σ1 = {0, 1, N}, σ2 = {0, 1,M} on Y and t(a) = r, t(b) = h. One may notice that t is fuzzy (1, 2)−gβ− conts, but not (1, 2)−gs− conts as t−1(M c) ≤ E ∈ δ1 but δ2 − s− cl(t−1(M c)) ≰ E. The example follow indicates that (1, 2)− gβ − conts ⇏ (1, 2)− gp− conts. Example 5. Suppose E,F,G, and H are fuzzy subgroups of X = {a, b}. We determine them as: E(a, b) = {0.5, 0.7} F (a, b) = {0.4, 0.6}, G(a, b) = {0.3, 0.4}, and H(a, b) = {0.6, 0.5}. Consider the fuzzy bitopology δ1 = {0, 1, E}, δ2 = {0, 1, F,G} on X. Suppose N , M are fuzzy subgroups of Y = {r, h} defined as: N(r, h) = {0.7, 0.5}, M(r, h) = {0.5, 0.5}. Consider the fuzzy bitopology σ1 = {0, 1, N}, σ2 = {0, 1,M} on Y and t(a) = r, t(b) = h. One may notice that t is fuzzy (1, 2)−gβ−conts, but not (1, 2)−gp−conts as t−1(M c) ≤ E ∈ δ1 but δ2 − p− cl(t−1(M c)) ≰ E. Theorem 3. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy δj − ψ − conts. Thus t is fuzzy (i, j)− gψ − conts. Proof. Suppose t is fuzzy δj − ψ − conts and U ∈ Fσj . Then t−1(U) ∈ FψC(X, δj), and hence t−1(U) is fuzzy (i, j)− gψ − cld of X. Therefore t is fuzzy (i, j)− gψ − conts. Remark 4. The inverse of the above theorem is incorrect. The next example is evidenced that: Suppose E, F are fuzzy subgroups of X = {a, b}. We determine them as: E(a, b) = {0.3, 0.4}, F (a, b) = {0.3, 0.2}. Consider the fuzzy bitopology δ1 = {0, 1, E}, δ2 = {0, 1, F} on X. Suppose N , M are fuzzy subgroups of Y = {r, h} defined as: N(r, h) = {0.6, 0.7}, M(r, h) = {0.3, 0.3}. Consider the fuzzy bitopology σ1 = {0, 1, N}, σ2 = {0, 1,M} on Y and t(a) = r, t(b) = h. One may notice that t is fuzzy (1, 2)− g− conts, so by Theorem 2, t is fuzzy (1, 2)− gα− conts but not δ2−α− conts since δ2−α− cl(t−1(M c)) ≰ t−1(M c). Corollary 2. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy δj − conts. Then t is (i, j) − gψ − conts. Theorem 4. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is injection mapping. Hence the next statements are equivalent: A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2619 (i) t((i, j)− gψ − cl(E)) ≤ σj − cl(t(E)), ∀E ∈ IX . (ii) (i, j)− gψ − cl(t−1(F )) ≤ t−1(σj − cl(F )), ∀F ∈ IY . (iii) t−1(σj − int(F )) ≤ (i, j)− gψ − int(t−1(F )),∀F ∈ IY . Proof. (i) → (ii) Suppose F ∈ IY . Then t−1(F ) ∈ IX by (i) we find t((i, j)− gψ− cl(t−1(F ))) ≤ σj − cl(t(t−1(F ))) ≤ σj − cl(F ). So, (i, j)− gψ − cl(t−1(F )) ≤ t−1(σj − cl(F )). (ii) → (iii) Suppose F ∈ IY . By (ii) (i, j) − gψ − cl(t−1(F )) ≤ t−1(σj − cl(E)). After that (t−1(σj − cl(E)))c ≤ ((i, j)− gψ − cl(t−1(E)))c, and hence t−1(σj − int(E)c) ≤ (i, j)− gψ − int(t−1(E)c). (iii) → (i) Suppose E ∈ IX . Then t(E) ∈ IY by (iii) t−1(σj − int(t(E)) ≤ (i, j) − gψ − int(t−1(t(E)). As t is injection then, t−1(σj − int(t(E)) ≤ (i, j) − gψ − int(E). After that (i, j)− gψ− cl((E)c) ≤ t−1(σj − cl(t(E)c). So, t((i, j)− gψ− cl((E)c)) ≤ t(t−1(σj − cl(t(E)c)) ≤ σj − cl(t(E)c). Theorem 5. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j) − gψ − conts. Hence t((i, j)− gψ − cl(E)) ≤ σj − cl(t(E)), for each E ∈ IX . Proof. Assume t is fuzzy (i, j)−gψ− conts and E ∈ IX . Then E ≤ t−1(σj− cl(t(E))). As σj − cl(t(E)) ∈ FC(Y, σj), t is fuzzy (i, j)− gψ− conts, thus t−1(σj − cl(t(E))) is fuzzy (i, j)−gψ−cld of X, so (i, j)−gψ−cl(E) ≤ t−1(σj−cl(t(E))). So, t((i, j)−gψ−cl(E)) ≤ σj − cl(t(E)). Remark 5. (1) If t is an injective and fuzzy (i, j) − gψ − conts mapping. Then any statement of Theorem 4 is hold in Theorem 5 since they are equivalent. (2) The converse of Theorem 5, Corollary 5 is incorrect because if we have a fuzzy set F ∈ Fσj , then by (ii) for example in Theorem 4 we find (i, j) − gψ − cl(t−1(F )) = t−1((F )) but t−1((F )) is not fuzzy (i, j)− gψ − cld group in X. Theorem 6. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2), g : (Y, σ1, σ2) → (Z, η1, η2) are fuzzy (i, j)− gψ − conts, hence g ◦ t : (X, δ1, δ2) → (Z, η1, η2) is not fuzzy (i, j)− gψ − conts. We show that if the functions t with g are fuzzy (i, j) − gα − conts by the coming example: Example 6. Suppose E,F , and G are fuzzy subgroups of X = {a, b} known as: E(a, b) = {0.3, 0.4}, F (a, b) = {0.4, 0.5}, G(a, b) = {0.4, 0.4}. Consider the fuzzy bitopol- ogy δ1 = {0, 1, E} also δ2 = {0, 1, F,G} on X. Suppose N with M are fuzzy sub- groups of Y = {r, h} defined as: N(r, h) = {0.5, 0.4}, M(r, h) = {0.5, 0.5}. Consider the fuzzy bitopology σ1 = {0, 1, N}, σ2 = {0, 1,M} on Y , and t(a) = r, t(b) = h. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2620 Suppose L and K are fuzzy subgroups of Z = {a, b} defined as: L(a, b) = {0.2, 0.3}, K(a, b) = {0.7, 0.5}. Consider the fuzzy bitopology η1 = {0, 1, L} and η2 = {0, 1,K} on Z, g : (Y, σ1, σ2) → (Z, η1, η2) such that g(a) = a, g(b) = b. one may notice that t, g are fuzzy (1, 2) − g − conts, and hence by Theorem 2 they are (i, j) − gα − conts, but g ◦ t is not fuzzy (1, 2)− gα− conts since (g ◦ t)−1(Kc) ≤ E ≤ δ1, but δ2 −α− cl((g ◦ t)−1(Kc)) ≰ E. Theorem 7. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is δj − ψ − conts, g : (Y, σ1, σ2) → (Z, η1, η2) is (i, j)−gψ−conts, and any fuzzy (i, j)−gψ−cld of Y is fuzzy open of (Y, σi). Thus g ◦ t : (X, δ1, δ2) → (Z, η1, η2) is (i, j)− gψ − conts. Proof. Assume W is fuzzy closed of (Z, ηj). As g is fuzzy (i, j) − gψ − conts, then g−1(W ) is fuzzy (i, j)− gψ − cld of Y . So by hypotheses g−1(W ) is fuzzy open of (Y, σi), so g−1(W ) is fuzzy δj −ψ− cld of (Y, σj). As t is fuzzy δj −ψ− conts, then t−1(g−1(W )) is fuzzy δj −ψ− cld of X, thus (g ◦ t)−1(W ) is fuzzy (i, j)− gψ− cld of X. Therefore g ◦ t is fuzzy (i, j)− gψ − conts. Theorem 8. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)−gψ−conts, g : (Y, σ1, σ2) → (Z, η1, η2) is σj − conts. Therefore g ◦ t : (X, δ1, δ2) → (Z, η1, η2) is (i, j)− gψ − conts. Proof. Assume W is fuzzy closed of (Z, ηj). As g is fuzzy σj − conts, then g−1(W ) is fuzzy closed of (Y, σj). As t is fuzzy (i, j) − gψ − conts, then t−1(g−1(W )) is fuzzy (i, j)− gψ − cld of X. Therefore g ◦ t is fuzzy (i, j)− gψ − conts. Theorem 9. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy δj − ψ − conts and δi − open mapping. Then any fuzzy (i, j)− gψ− cld group F of Y , t−1(F ) is (i, j)− gψ− cld of X. Proof. Assume F is fuzzy (i, j) − gψ − cld group of Y , W is fuzzy open of (X, δi) including t−1(F ). Since t is fuzzy δi − open, then t(W ) is fuzzy open of (Y, σi). As F ≤ t(W ), F is fuzzy (i, j) − gψ − cld of Y , then σj − ψ − cl(F ) ≤ t(W ) which implies t−1(σj − ψ − cl(F )) ≤ W . Since t is fuzzy δj − ψ − conts, hence t−1(σj − ψ − cl(F )) = δj − ψ − cl(t−1(σjψ − cl(F ))). Then δj − ψ − cl(t−1(F )) ≤ t−1(σj − ψ − cl(F )) ≤W . So, t−1(F ) is fuzzy (i, j)− gψ − cld of X. 4. Types of Fuzzy Generalized Strongly Continuity and Irresolute Mapping In the coming part, we define fuzzy generalized strong continuity and irresolute map- ping of generalized closed sets and we denote by (i, j) − gψ−strongly conts, and (i, j) − gψ−irresolute respectively after that we study some properties and theorem for them and presented counter examples too. Definition 10. A function t : (X, δ1, δ2) → (Y, σ1, σ2) is claimed: (1) Fuzzy (i, j)−generalizedψ−strongly continuous (briefly, (i, j)−gψ−strongly conts) when t−1(W ) is fuzzy δj − closed group of X for all W is (i, j)− gψ − cld of Y . A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2621 (2) Fuzzy (i, j) − generalizedψ − irresolute mapping (briefly, (i, j) − gψ − irresolute) when t−1(W ) is fuzzy (i, j)− gψ − cld of X for all W is (i, j)− gψ − cld of Y . Theorem 10. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Thus, the next claims are accurate: (1) If t is fuzzy (i, j)− gψ − strongly conts, thus it is δj − conts. (2) If t is (i, j)− gψ − strongly conts, thus it is (i, j)− gψ − irresolute. (3) If t is (i, j)− gψ − irresolute, thus it is (i, j)− gψ − conts. Proof. (1) Assume t is (i, j)− gψ− strongly conts, W ∈ Fσj . Since W is fuzzy closed of (Y, σj), then W is (i, j) − gψ − cld group of Y . As t is fuzzy (i, j) − gψ − strongly conts, thus t−1(W ) ∈ Fδj . Therefore t is δj − conts. (2) Assume t is (i, j) − gψ − strongly conts and W is (i, j) − gψ − cld group of Y . Thus t−1(W ) ∈ Fδj , so t−1(W ) is (i, j) − gψ − cld group of X. Consequently, t is (i, j)− gψ − irresolute mapping. (3) Assume t is (i, j) − gψ − irresolute, W ∈ Fσj . Since W is fuzzy closed of (Y, σj), then W is (i, j) − gψ − cld of Y . As t is (i, j) − gψ − irresolute, thus t−1(W ) is (i, j)− gψ − cld of X. So, t is (i, j)− gψ − conts. Remark 6. The next diagram explaining the relation in each statements in the above theorem: Figure 2: Presents the relationships via all varieties of fuzzy (i, j)− gψ −mappings. The coming examples clear the opposite implications of Figure (2) are generally not true and also clear that the fuzzy δj − conts and (i, j)− gψ − irresolute are independent as we explain that for type ψ is fuzzy α−open. Example 7. Suppose E,B,G, and H are subgroups of X = {a, b} defined as: E(a, b) = {0.5, 0.4}, F (a, b) = {0.7, 0.5}, G(a, b) = {0.4, 0.3}. Consider the fuzzy bitopol- ogy δ1 = {0, 1, E}, δ2 = {0, 1, F,G} on X. Suppose N ,M are fuzzy subgroups of Y = {r, h} defined as follows: N(r, h) = {0.3, 0.1}, M(r, h) = {0.7, 0.6}. Consider the fuzzy bitopol- ogy σ1 = {0, 1, N}, σ2 = {0, 1,M} on Y and t(a) = r, t(b) = h. One may notice that M c is fuzzy σ2 − closed of Y and t−1(M c) is fuzzy (1, 2) − gα − cld of X, so t is (1, 2) − gα − conts, but not δ2 − conts since δ2 − cl(t−1(M c)) not closed of X. Also, we find t is (i, j)− gψ − irresolute but not fuzzy (1, 2)− gα− strongly conts since M c is fuzzy (1, 2)− gα− cld of Y but t−1 is not closed of X. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2622 Theorem 11. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)−gψ−irresolute mapping, with all (i, j)− gψ− cld of X is fuzzy open in (X, δi). Hence t is δj −ψ− conts mapping. Proof. AssumeW ∈ Fσj . ThenW is (i, j)−gψ−cld of Y . As t is (i, j)−gψ−irresolute, then t−1(W ) is (i, j) − gψ − cld of X. Thus by hypotheses t−1(W ) ∈ FO(X, δi), so t−1(W ) ∈ FψC(X, δj). Theretore t is fuzzy δj − ψ − conts mapping. Theorem 12. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy δj−ψ−irresolute mapping, and every fuzzy (i, j)−gψ−cld of Y is fuzzy open of (Y, σi). Hence t is (i, j)−gψ− irresolute mapping. Proof. Suppose W is (i, j) − gψ − cld group of Y . Then by hypotheses W ∈ (X, δi), and hence W ∈ FψC(Y, σj). As t is δj − ψ − irresolute, then t−1(W ) ∈ FψC(X, δj), and hence t−1(W ) is fuzzy (i, j) − gψ − cld of X. Hence, t is fuzzy (i, j) − gψ − irresolute mapping. Theorem 13. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Thus, the next claims are accurate: (1) If t is (i, j)− gβ− strongly conts, so it is (i, j)− gs− strongly conts also (i, j)− gp− strongly conts. (2) If t is (i, j)− gs− strongly conts or (i, j)− gp− strongly conts, so it is (i, j)− gα− strongly conts. (3) If t is (i, j)− gα− strongly conts, so it is (i, j)− g − strongly conts. Proof. It is clear by uses Theorems in [3]. Remark 7. The following diagram explaining the relation in each statements in the above theorem: Figure 3: Presents the relationships via all varieties of fuzzy (i, j)− gψ − strongly conts. The converses of the above relations are not valid in general and this is based on the relationships between the (i, j) − gψ − cld groups that were explained with examples in Reference No [3]. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2623 Theorem 14. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)−gψ−irresolute mapping. After that t((i, j)− gψ − cl(E)) ≤ σj − ψ − cl(t(E)), ∀E ∈ IX . Proof. Assume t is (i, j)−gψ−irresolute, E ∈ IX . Thus E ≤ t−1(σj−ψ−cl(t(E))), and hence σj−ψ− cl(t(E)) is fuzzy (i, j)−gψ− cld of Y . As t is fuzzy (i, j)−gψ− irresolute, so t−1(σj − ψ − cl(t(E))) is fuzzy (i, j) − gψ − cld of X, thus (i, j) − gψ − cl(E) ≤ t−1(σj − ψ − cl(t(E))). Therefore t((i, j)− gψ − cl(E)) ≤ σj − ψ − cl(t(E)). Corollary 3. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j) − gψ − strongly conts mapping. Hence t((i, j)− gψ − cl(E)) ≤ σj − ψ − cl(t(E)), ∀E ∈ IX . Theorem 15. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)− gψ − irresolute (resp, (i, j)−gψ−strongly conts) mapping. Thus t((i, j)−gψ−cl(E)) ≤ σj−cl(t(E)), ∀E ∈ IX . Proof. It is clear from Theorem 10 and Theorem 5. Remark 8. From Remark 5 and Theorem 10 we find when t : (X, δ1, δ2) → (Y, σ1, σ2) is an injective and fuzzy (i, j)−gψ− irresolute (resp, (i, j)−gψ−strongly conts) mapping. Hence any statement of Theorem 4 is hold in Theorem 15. Theorem 16. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2), g : (Y, σ1, σ2) → (Z, η1, η2). Hence the next cliams are true: (1) g ◦ t is fuzzy (i, j)− gψ − strongly conts if t, g are fuzzy (i, j)− gψ − strongly conts. (2) g ◦ t is fuzzy (i, j)− gψ − irresolute if t, g are fuzzy (i, j)− gψ − irresolute. (3) g ◦ t is fuzzy (i, j)− gψ− conts if g is (i, j)− gψ− conts, t is (i, j)− gψ− irresolute. Proof. (1) Assume W is fuzzy (i, j) − gψ − cld of Z. As g is fuzzy (i, j) − gψ − strongly conts, hence g−1(W ) is fuzzy closed of Y , so g−1(W ) is (i, j)− gψ − cld group of Y . As t is (i, j) − gψ − strongly conts, then t−1(g−1(W )) = (g ◦ t)−1(W ) is fuzzy closed of X. Therefore g ◦ t is fuzzy (i, j)− gψ − strongly conts. (2) Assume W is fuzzy (i, j) − gψ − cld of Z. As g is fuzzy (i, j) − gψ − irresolute, hence g−1(W ) is fuzzy (i, j) − gψ − cld of Y . As t is (i, j) − gψ − irresolute, then t−1(g−1(W )) = (g ◦ t)−1(W ) is fuzzy (i, j)− gψ − cld of X. Therefore g ◦ t is fuzzy (i, j)− gψ − irresolute. (3) Suppose W ∈ Fηj . Since g is fuzzy (i, j) − gψ − conts, then g−1(W ) is fuzzy (i, j) − gψ−cld of Y . As t is fuzzy (i, j)−gψ− irresolute, thus t−1(g−1(W )) = (g◦ t)−1(W ) is fuzzy (i, j)− gψ − cld of X. Therefore g ◦ t is fuzzy (i, j)− gψ − conts. Theorem 17. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy δj − ψ − conts and δi − open mapping (resp, δi − closed). Hence t is fuzzy (i, j)− gψ − irresolute mapping. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2624 Proof. It is clear from Theorem 9 and Definition 10. Theorem 18. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Consequently, the claims below are equivalent: (i) t is fuzzy (i, j)− gψ − irresolute mapping. (ii) The converse of all fuzzy (i, j)− gψ − open of Y is (i, j)− gψ − open of X. Proof. It’s clear by taking the complement of Definition 10. Theorem 19. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Consequently, the next claims are equivalent: (i) t is fuzzy (i, j)− gψ − strongly conts. (ii) The inverse of every fuzzy (i, j)− gψ − open group of Y is δj − open group of X. Proof. It is clear by taking the complement of Definition 10. Theorem 20. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2), g : (Y, σ1, σ2) → (Z, η1, η2). Hence the next claims are true: (1) g ◦ t is fuzzy δj − conts when g is fuzzy (i, j) − gψ − conts and t is (i, j) − gψ − strongly conts. (2) g ◦ t is fuzzy (i, j)− gψ − strongly conts when g is (i, j)− gψ − strongly conts and t is δj − conts. (3) g ◦ t is fuzzy (i, j) − gψ − strongly conts when g is (i, j) − gψ − irresolute, t is (i, j)− gψ − strongly conts. Proof. (1) Assume W ∈ Fηj . As g is (i, j)− gψ − conts, thus g−1(W ) is (i, j)− gψ − cld group of Y . Since t is fuzzy (i, j)− gψ− strongly conts, then t−1(g−1(W )) = (g ◦ t)−1(W ) is fuzzy δj − closed of X. Therefore g ◦ t is fuzzy δj − conts. (2) AssumeW is fuzzy (i, j)−gψ−cld group of Z. As g is fuzzy (i, j)−gψ−strongly conts, thus g−1(W ) is fuzzy closed (Y, σj). As t is fuzzy δj − conts, then t−1(g−1(W )) = (g ◦ t)−1(W ) ∈ Fδj . So, g ◦ t is fuzzy (i, j)− gψ − strongly conts. (3) AssumeW is fuzzy (i, j)−gψ−cld group of Z. As g is fuzzy (i, j)−gψ−irresolute, thus g−1(W ) is fuzzy (i, j)−gψ−cld group of Y . As t is fuzzy (i, j)−gψ−strongly conts, hence t−1(g−1(W )) = (g◦t)−1(W ) ∈ Fδj . So, g◦t is fuzzy (i, j)−gψ−strongly conts. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2625 5. Types of Fuzzy Generalized Open and Closed Mappings In the third part, we introduce some concepts for fuzzy (i, j)−generalized ψ−open and closed mapping then we study some properties and important theorems. Definition 11. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Thus t is referred to as: (1) fuzzy (i, j)− generalized ψ − open mapping (briefly, (i, j)− gψ − open) when t(V ) is fuzzy (i, j)− gψ − open in Y for any V is (i, j)− gψ − open of X. (2) fuzzy (i, j)− generalized ψ− closed mapping (briefly, (i, j)− gψ− closed) when t(V ) is fuzzy (i, j)− gψ − cld of Y for any V is (i, j)− gψ − cld of X. Theorem 21. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy σj−open function. Thus t(K) is fuzzy (i, j)− gψ − open in Y for any K ∈ (x, δj). Proof. Assume t is σj − open, K ∈ Oδj . So t(K) ∈ (Y, σj), and hence by relation via open group and (i, j) − gψ − open group which was clear in refrence [3], we find t(K) is fuzzy (i, j)− gψ − open in Y . Theorem 22. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2), g : (Y, σ1, σ2) → (Z, η1, η2). Then the claims below are true: (1) g◦t is fuzzy (i, j)−gψ−open (resp, (i, j)−gψ−closed) if t, g are (i, j)−gψ−open (resp, (i, j)− gψ − closed) mapping. (2) t is fuzzy (i, j)−gψ−open (resp, (i, j)−gψ− closed) if g ◦ t is (i, j)−gψ−open (resp, (i, j)− gψ − closed), g is (i, j)− gψ − irresolute and injective mapping. (3) t is fuzzy δj−open (resp, δj−closed) if g◦t is (i, j)−gψ−open (resp, (i, j)−gψ−closed), g is (i, j)− gψ − strongly conts and injective mapping. (4) g is fuzzy (i, j)−gψ−open (resp, (i, j)−gψ−closed) if g ◦ t is (i, j)−gψ−open (resp, (i, j)− gψ − closed), t is (i, j)− gψ − irresolute and surjective mapping. (5) g is fuzzy σj − open (resp, σj − closed) if g ◦ t is δj − open (resp, δj − closed), t is (i, j)− gψ − strongly conts and surjective mapping. Proof. (1) Assume K is fuzzy (i, j)− gψ− open group of X. As t is (i, j)− gψ− open mapping, thus t(K) is (i, j)− gψ− open group of Y . As g is (i, j)− gψ− open mapping, hence g(t(K)) is (i, j)− gψ− open group of Z. So (g ◦ t)(K) is (i, j)− gψ− open of Z. So, g ◦ t is (i, j)− gψ − open mapping. (2) Assume K is (i, j)−gψ−open group of X. As g ◦ t is (i, j)−gψ−open mapping, then g(t(K)) is (i, j) − gψ − open group of Z. As g is (i, j) − gψ − irresolute mapping, injective, then t(K) is (i, j)− gψ− open group of Y . So, t is fuzzy (i, j)− gψ− open mapping. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2626 (3) Assume K ∈ δj . After that K considers (i, j) − gψ − open group of X. As g ◦ t is (i, j) − gψ − open mapping, thus g(t(K)) is (i, j) − gψ − open group of Z. As g is (i, j) − gψ − stongly conts mapping with injective, then t(K) ∈ σj . Therefore t is fuzzy δj − open mapping. (4) Assume W is fuzzy (i, j) − gψ − open group of Y .As t is (i, j) − gψ − irresolute mapping, thus t−1(W ) is (i, j)− gψ− open group of X. As g ◦ t is (i, j)− gψ− open mapping, t is surjective, then g(W ) is (i, j)− gψ − open group of Z. So, g is fuzzy (i, j)− gψ − open mapping. (5) Assume W ∈ σj . Then W is (i, j) − gψ − open group of Y . As t is (i, j) − gψ − stongly conts mapping, then t−1(W ) ∈ δj . As g ◦ t is δj − open, with t is surjective, thus g(W ) ∈ ηj . So, g is σj − open mapping Corollary 4. (1) Suppose t : (X, δ1, δ2) → (Y, σ1, σ2), g : (Y, σ1, σ2) → (Z, η1, η2) is (i, j)− gψ − strongly conts, injective mapping, also g ◦ t is (i, j)− gψ − open (resp, (i, j)−gψ−closed). Thus, t is (i, j)−gψ−open (resp, (i, j)−gψ−closed) mapping. (2) Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is (i, j)− gψ− strongly conts, surjective function, g : (Y, σ1, σ2) → (Z, η1, η2), also g ◦ t is (i, j)− gψ− open (resp, (i, j)− gψ− closed). Thus g is (i, j)− gψ − open (resp, (i, j)− gψ − closed) mapping. (3) Suppose t : (X, δ1, δ2) → (Y, σ1, σ2), g : (Y, σ1, σ2) → (Z, η1, η2) is (i, j) − gβ − strongly conts, injective mapping, also g ◦ t is (i, j)− gψ − open (resp, (i, j)− gψ − closed). Thus t is δj − open (resp, δj − closed) mapping. Theorem 23. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)−gψ−open (resp, (i, j)− gψ − closed) mapping. Thus all W is fuzzy subgroup of Y , and K ∈ Fδj (resp,K ∈ δj) including t−1(W ), ∃ W is fuzzy (i, j)− gψ − cld (resp, (i, j)− gψ − open) of Y including W as t−1(W ) ≤ K. Proof. Assume t is (i, j)− gψ − open, K ∈ Fδj as t−1(W ) ≤ K, as W ∈ IY . So Kc ≤ (t−1(W ))c = t−1(W c). Since t is fuzzy (i, j)−gψ−open, also Kc is fuzzy (i, j)−gψ−open group of X, hence t(Kc) is fuzzy (i, j) − gψ − open of Y and t(Kc) ≤ W c, and hence W ≤ (t(Kc))c if we chose W = (t(Kc))c, thus ∃W is fuzzy (i, j) − gψ − cld group of Y including W as t−1(W ) = K. Theorem 24. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is bijective mapping. Hence the next claims are equivalent: (i) t is fuzzy (i, j)− gψ − open mapping. (ii) t is fuzzy (i, j)− gψ − closed mapping. Proof. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2627 (i) → (ii) Suppose t is (i, j)− gψ− open mapping, K is (i, j)− gψ− cld of X. Thus Kc is (i, j)−gψ−open of X. As t is (i, j)−gψ−open mapping, so t(Kc) is (i, j)−gψ−open of Y . As t is bijective, then t(X) = Y , hence Y − t(K) = (t(K))c is (i, j)−gψ−open of Y , then t(K) is (i, j)− gψ − cld of Y . So, t is (i, j)− gψ − closed mapping. (ii) → (i) Suppose t is (i, j) − gψ − closed mapping with K is (i, j) − gψ − open of X. Thus Kc is (i, j)− gψ − cld of X. As t is (i, j)− gψ − closed mapping, so t(Kc) is (i, j)− gψ− cld of Y . As t is bijective, so t(X) = Y , and hence Y − t(K) = (t(K))c is (i, j)−gψ−cld of Y , then t(K) is (i, j)−gψ−open of Y . So, t is (i, j)−gψ−open mapping. Theorem 25. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)−gψ−open (resp, (i, j)− gψ− closed) mapping. Thus for any R ∈ IX , t(δj −ψ− int(R)) ≤ (i, j)− gψ− int(t(R)). Proof. Suppose t is (i, j) − gψ − open mapping with R is subgroup of X. Since t(δj − ψ − int(R)) ≤ t(R) and δj − ψ − int(R) considers (i, j) − gψ − open group of X. Then t(δj − ψ − int(R)) ≤ (i, j)− gψ − int(t(R)). 6. Fuzzy Generalized Homomorphism Mapping Finally, we investigate some theorems for the fuzzy generalized homomorphism and we denote by gψ−homomorphism. Definition 12. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2). Hence t is known as fuzzy gψ−homomorphism if and only if the claims below are true: (1) t is bijective. (2) t is fuzzy (i, j)− gψ − conts. (3) t−1 is fuzzy (i, j)− gψ − conts. Remark 9. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2), g : (Y, σ1, σ2) → (Z, η1, η2) both of them are fuzzy gψ−homomorphism mapping. Then g◦t is not fuzzy gψ−homomorphism because g ◦ t is not fuzzy (i, j)− gψ − conts. Refer to Theorem 6. Theorem 26. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy gψ−homomorphism, fuzzy (i, j)−gψ−irresolute mapping, with g : (Y, σ1, σ2) → (Z, η1, η2) is fuzzy gψ−homomorphism mapping. Thus g ◦ t is fuzzy gψ−homomorphism. Proof. Suppose R ∈ Fηj . Hence g−1(R) is (i, j) − gψ − cld of Y . As t is fuzzy (i, j)− gψ − irresolute mapping, then t−1(g−1(R)) is (i, j)− gψ − cld of X. So (g ◦ t)(R) is fuzzy (i, j)− gψ − conts. Assume R is (i, j) − gψ − open of X. As t−1 is (i, j) − gψ − conts, then (t−1)−1(R) = t(R) ∈ Fσj , and hence t(R) is (i, j)− gψ − cld of Y . Then (g−1)−1t(R) = g(t(R)) ∈ Fηj , A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2628 thus ((g ◦ t)−1)−1(R) = (g ◦ t)(R) = g(t(R)) is fuzzy closed of (Z, ηj). So (g ◦ t)−1 is tuzzy (i, j)− gψ − conts. Then, assume (g ◦ t)(R1) = (g ◦ t)(R2). Then g(t(R1)) = g(t(R2)), as t is injective, then g(R1) = g(R2). As g is injective, then R1 = R2. Therefore g ◦ t is injective mapping. Also, assume c ∈ Z. As g is surjective, hence ∃ b ∈ Y such that g(b) = c. As t is also surjective, hence for any b ∈ Y ∃ a ∈ X as t(a) = b, and hence for any c ∈ Z ∃ a ∈ X such that (g ◦ t)(a) = g(t(a)) = g(b) = c. Thus g ◦ t is surjection. Hence g ◦ t is bijective. So, g ◦ t is fuzzy gψ−homomorphism. Corollary 5. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy gψ−homomorphism, (i, j) − gψ − strongly conts, g : (Y, σ1, σ2) → (Z, η1, η2) is gψ−homomorphism mapping. Then g ◦ t is fuzzy gψ−homomorphism. Proof. Since t : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j) − gψ − strongly conts, so by Theorem 10 t is fuzzy (i, j)− gψ − irresolute mapping, and hence by Theorem 26 we get the desired. Theorem 27. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is bijective function, t−1 is (i, j)−gψ− conts mapping. Then t is (i, j)− gψ − open (resp, (i, j)− gψ − closed) mapping. Proof. Assume t−1 is fuzzy (i, j) − gψ − conts, R is fuzzy (i, j) − gψ − open of X. Then t−1(R) ∈ σj , hence t −1(R) is fuzzy (i, j) − gψ − open of Y . As t is bijection, then (t−1)−1(R) = t(R). So t(R) is fuzzy (i, j)−gψ−open of Y . So, t is fuzzy (i, j)−gψ−open mapping. Corollary 6. Suppose t : (X, δ1, δ2) → (Y, σ1, σ2) is bijective function, gψ−homomorphism. Then t is (i, j)− gψ − open (resp, (i, j)− gψ − closed) mapping. Proof. Assume t is fuzzy gψ−homomorphism. Then by Definition 12 we find t−1 is (i, j)− gψ − conts mapping, hence by Theorem 27 achieved what we want to be proved. In the following table, we summarize the composition process among all types of func- tions: (i, j)−gψ−conts, (i, j)−gψ−stronglyconts, (i, j)−gψ−irresolute, (i, j)−gψ−open, δj −ψ− conts, δj − open, σj − conts, and σj − open, where zero indicates that there is no result of the outcome while 1 indicates that the composition process is possible and has a result. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (4) (2023), 2613-2631 2629 Remark 10. (1) To clarify further, for example, if g is (i, j)−gψ−open and t is (i, j)− gψ − irresolute, we notice from the table that the result of the composition process is equal to 0 because of the difference in the effect of domains, as t is moved from Y to X, but g is moved from Y to Z, there is no connection between X and Z, so the result equals 0, while we find that if g, t are (i, j)− gψ − irresolute, thus g ◦ t = 1, because g ◦ t has a result and its (i, j)− gψ − irresolute too, see Theorem 16. (2) We would like to note that the composition process collecting t, g is not equal to the process of collecting g, t. i.e. g ◦ t ̸= t ◦ g. (3) A detailed explanation of the resultant of the composition, t if δj − open and g if (i, j)− gψ − open. The reslut g ◦ t = 0, but (g ◦ t)(K) is (i, j)− gψ − open group of Z, when K is open group of X. 7. 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